Commutative algebra in algebraic geometry and algebraic combinatorics

Texas A&M University, College Station TX

Investigators

Abstract

Many constraints governing real-world problems, such as those arising in robotics or statistics, are described (or approximated) by solutions to polynomial equations. The set of solutions to a polynomial equation makes up a geometric object in multi-dimensional space (e.g., the set of solutions to the equation y=x^2 is a parabola in 2-space). The intersection of several of these geometric objects is made up of the points that are solutions to all of polynomial equations cutting out those geometric objects. When such an intersection consists of finitely many points, it is natural to ask how many points. Questions of this nature have been of broad interest within algebraic geometry and algebraic combinatorics since at least the time of Hermann Schubert (1848-1911), whose methodology for solving these problems forms the basis of what is now called Schubert calculus. The PI will apply techniques from commutative algebra to solve problems in modern-day Schubert calculus. Through her research program, she will train graduate students. She will also promote gender equity in the mathematical sciences by taking on leadership roles in conferences and research communities for women and non-binary mathematicians working in commutative algebra. The primary theme of this research project is the use of combinatorial models to understand affine varieties and the algebro-geometric significance of these combinatorial models. The PI will first focus on Schubert varieties in the complete flag variety and the Groebner geometry of several classes of related affine varieties. A motivating problem is to understand the Schubert structure constants. Another family of problems concerns the connection between Gorenstein liaison and Groebner degenerations. The PI will use this connection to study a range of combinatorially-natural varieties. She will also further the understanding of Gorenstein liaison itself. A link between these two focuses is the family of alternating sign matrix (ASM) varieties, which are generalizations of matrix Schubert varieties. Understanding when and why ASM varieties fail to exhibit various “niceness” properties of matrix Schubert varieties will shed light on which features of matrix Schubert varieties are essential to their good behavior. It will also facilitate the use of ASM varieties in testing an open question in Gorenstein liaison. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.